Cutting to the chase: the plot of the various aggregation models follows. Just under the plot heading you can see the end-point for each of the models (as at 22 April 2018). As usual, the input data for these models comes from the Wikipedia page on the Next Australian Federal Election.

The estimated house biases (the first six columns expressed in pro-Coalition percentage points) for each model (the rows) are in the next table. Note, these are relative house effects as the rows have been constrained to sum to zero. The "Iter" column is the number of iterations taken to produce this estimate. The "Sum Errors Squared" column is the sum of the errors squared, noting that within the model these are calculated from proportions (between 0 and 1) and not percentage points (between 0 and 100).

Essential | Ipsos | Newspoll | ReachTEL | Roy Morgan | YouGov | Iter | Sum Errors Squared | |
---|---|---|---|---|---|---|---|---|

Model | ||||||||

HMA-181 | -0.845100 | -0.568599 | -0.615695 | -0.408452 | 0.278507 | 2.159338 | 13 | 0.008061 |

HMA-365 | -0.832409 | -0.485705 | -0.589923 | -0.401716 | 0.150488 | 2.159266 | 12 | 0.008920 |

LOWESS-91 | -0.818754 | -0.554349 | -0.604249 | -0.403321 | 0.216127 | 2.164546 | 13 | 0.008110 |

LOWESS-181 | -0.826693 | -0.475102 | -0.577678 | -0.413161 | 0.169364 | 2.123270 | 12 | 0.009222 |

This compares well with the Hierarchical Bayesian model:

The updated code follows.

# PYHTON: iterative data fusion: # using Henderson Moving Averages (HMA) # and Locally Weighted Scatterplot Smoothing (LOWESS) # with a sum-to-zero adjustment for House biases import pandas as pd import numpy as np from numpy import dot import matplotlib.pyplot as plt import matplotlib.dates as mdates import statsmodels.api as sm lowess = sm.nonparametric.lowess import sys sys.path.append( '../bin' ) from mg_plot import mg_min_max_end from Henderson import Henderson plt.style.use('../bin/markgraph.mplstyle') # --- key constants HMA_PERIODS = [181, 365] # days LOWESS_PERIODS = [91, 181] # days MODALITIES = ['HMA', 'LOWESS'] graph_dir = './Graphs/' graph_leader = 'FUSION-' intermediate_data_dir = "./Intermediate/" # --- Functions def note_house_effects(effects, Houses, mode, period, iter_count, current_sum): ''' For each iteration we record the results. This function compiles the results into a single row DataFrame, which will be appended to the iteration history DataFrame effects: is a column vector of house effects that was applied Houses: is a list of Houses mode: is a mode in 'HMA' or 'LOWESS' iter_count: is the iteration count as an integer current_sum: is the error squared sum as a float returns: a Pandas DataFrame with one row ''' house_effects = pd.DataFrame([effects.T[0]], columns=Houses, index=[iter_count]) house_effects['Iterations'] = iter_count house_effects['Model'] = '{}-{}'.format(mode, period) house_effects['Error Sq Sum'] = [current_sum] house_effects['Effects Sq Sum'] = dot(effects.T, effects)[0] return house_effects def interpolate_and_smooth(x): ''' interpolate missing data in a series then apply a 21-term moving average - with smaller moving averages at the end of the series. x: a Pandas Series returns: a Pandas Series ''' # set up smoothers - note centre weighting in 7- and 21-term MA smoother21 = np.array([1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,2,2,1,1]) smoother21 = smoother21 / np.sum(smoother21) smoother7 = np.array([1,2,3,3,3,2,1]) # 7-term MA smoother7 = smoother7 / np.sum(smoother7) smoother3 = np.array([1,1,1]) smoother3 = smoother3 / np.sum(smoother3) x = x.interpolate() # calculate smoothers s21 = x.rolling(window=len(smoother21), min_periods=len(smoother21), center=True).apply(func=lambda x: (x * smoother21).sum()) s7 = x.rolling(window=len(smoother7), min_periods=len(smoother7), center=True).apply(func=lambda x: (x * smoother7).sum()) s3 = x.rolling(window=len(smoother3), min_periods=len(smoother3), center=True).apply(func=lambda x: (x * smoother3).sum()) # fix the end data from smoothed to less smoothed to unsmoothed s = s7.where(s21.isnull(), other=s21) s = s3.where(s.isnull(), other=s) s = x.where(s.isnull(), other=s) return(s) def estimate_hidden_states(ydf, mode, period, n_days): ''' This function takes the house-effect adjusted y values and estimates a hidden vote share for each day under analysis using moving averages to give a smooth result. ydf: is a DataFram of y values, with cols: Day and adjusted_y mode: is a MODALITY string - either 'HMA' or 'LOWESS' period: in days - the span for the moving average returns: a pandas Series indexed by days ''' # --- plot known data points and interpolate the in-between days # where more than one poll on a day, average those polls. hidden_state = pd.Series(np.array([np.nan] * n_days)) for day in ydf['Day'].unique(): result = ydf[ydf['Day'] == day] hidden_state[day] = result['adjusted_y'].mean() # --- apply the HMA or LOWESS smoothing if mode == 'HMA': # HMA requires data points for all x values # therefore we interpolate first hidden_state = interpolate_and_smooth(hidden_state) hidden_state = Henderson(hidden_state, period) elif mode == 'LOWESS': # LOWESS does not require data points for all x values hidden_state = pd.Series( lowess(hidden_state, hidden_state.index, frac=period/n_days, return_sorted=False)) hidden_state = interpolate_and_smooth(hidden_state) else: assert(False) return hidden_state def calculate_house_effects(Houses, ydf, hidden_state): ''' For a curve generated by the estimate_hidden_states function, calculate the zero-sum constrained house effects for each pollster Houses: is a list of pollsters ydf: is a pandas DataFrame of y values/attributes, with columns y, Day and Firm returns: a column vector of house effects ''' new_effects = [] for h in Houses: count = 0 sum = 0 result = ydf[ydf['Firm'] == h] for index, row in result.iterrows(): sum += hidden_state[row['Day']] - row['y'] count += 1.0 new_effects.append(sum / count) new_effects = pd.Series(new_effects) new_effects = new_effects - new_effects.mean() # sum to zero effects = new_effects.values effects.shape = len(effects), 1 # it's a column vector return effects def sum_error_squared(hidden_state, ydf): ''' For a curve generated by the estimate_hidden_states function, calculate the sum of the errors-squared for the y observations hidden_state: a pandas Series indexed by days ydf: is a pandas DataFrame of y values/attributes, with columns Day and adjusted_y returns: a float ''' dayValue = hidden_state[ydf['Day']] dayValue.index = ydf.index e_row = (dayValue - ydf['adjusted_y']).values # row vector return dot(e_row, e_row.T) def get_minima(history): ''' Return the minimum sum of errors-squared from the iteration history DataFrane history: pandas DataFrame of all iterations to now returns: minimum value for the sum of errors squared ''' return history['Error Sq Sum'].min() def get_details(search, history, Houses): ''' Find the details in the iteration history DataFrame for a specific search value. The search term is the value of sum errors squared being sought (typically a minimum) search: the value of sum errors squared being sought history: pandas DataFrame of all iterations to now Houses: is a list of pollsters returns: (iter_num, effects) - found selected effects in history ''' effects = history[history['Error Sq Sum'] == search][Houses].T.values effects.shape = (len(effects), 1) # effects is a column vector iter_num = history[history['Error Sq Sum'] == search]['Iterations'] return (iter_num, effects) def curve_fit(Houses, H, mode, period, ydf, n_days): ''' Iteratively fit curves to the data, then adjust the data to better reflect the house effects for each pollster. Stop when the changes being made become minimal. Houses: is a list of Houses H: is a House Effects dummy var matrix mode: is a MODALITY string - either 'HMA' or 'LOWESS' - which reflects the type of curve we will fit period: in days - the span for the moving average ydf: pandas DataFrame of y variables with cols 'y', 'Day' 'Firm' n_days: number of days under analysis returns: (iter_count, history, y) ''' # --- initialisation regardless of mode effects = np.zeros(shape=(len(Houses), 1)) # start at zero history = pd.DataFrame() previous_sum = np.inf y = ydf['y'].values y.shape = (len(y), 1) # column vector iter_count = 0; # --- iterative fitting process ... # note: this is only a quick and dirty approximation print('--> About to iterate: {}-{}'.format(mode, period)) while True: iter_count += 1 # --- calculate new hidden states, # update estimate of house effects # and calculate error squared ydf['adjusted_y'] = y + dot(H, effects) # matrix arithmetic hidden_state = estimate_hidden_states(ydf, mode, period, n_days) effects = calculate_house_effects(Houses, ydf, hidden_state) current_sum = sum_error_squared(hidden_state, ydf) if iter_count > 1: minima = get_minima(history) else: minima = np.inf # Note: minima does not include current_sum # --- remember where we have been - puts current_sum into history house_effects = note_house_effects(effects, Houses, mode, period, iter_count, current_sum) history = history.append(house_effects) print('--\n', house_effects) # --- exit when we are no longer making much difference margin = 0.000000000001 if np.abs(current_sum - minima) < margin or np.abs( current_sum - previous_sum) < margin: # near enough to a minima break # --- end loop tidy-ups previous_sum = current_sum # --- exit return (iter_count, history, y) def chart(sdf=None, title=None, y_label=None, annotation='', file_prefix=None): ax = sdf.plot() ax.set_title(title) ax.set_xlabel('') ax.set_ylabel(y_label) ax.xaxis.set_major_formatter(mdates.DateFormatter('%b\n%y')) ax.text(0.01, 0.99, annotation, ha='left', va='top', fontsize='xx-small', color='#333333', transform = ax.transAxes) fig = ax.figure fig.set_size_inches(8, 4) fig.tight_layout(pad=1) fig.text(0.99, 0.01, 'marktheballot.blogspot.com.au', ha='right', va='bottom', fontsize='x-small', fontstyle='italic', color='#999999') fig.savefig(graph_dir+file_prefix+'.png', dpi=125) return (fig, ax) # --- collect the model data # the XL data file was extracted from the Wikipedia # page on the next Australian Federal Election workbook = pd.ExcelFile('./Data/poll-data.xlsx') df = workbook.parse('Data') # drop pre-2016 election data df['MidDate'] = [pd.Period(date, freq='D') for date in df['MidDate']] df = df[df['MidDate'] > pd.Period('2016-07-04', freq='D')] # convert dates to days from start start = df['MidDate'].min() # day zero df['Day'] = df['MidDate'] - start # day number for each poll n_days = df['Day'].max() + 1 # treat later Newspoll as a seperate series # for change to one nation treatment df['Firm'] = df['Firm'].where((df['MidDate'] < pd.Period('2017-12-01', freq='D')) | (df['Firm'] != 'Newspoll'), other='Newspoll2') df = df.sort_values(by='Day') df.index = range(len(df)) # reindex, just to be sure # --- do for a number of different HMAs and LOWESS functions Adjustments = pd.DataFrame() Hidden_States = pd.DataFrame() for mode in MODALITIES: if mode == 'HMA': PERIODS = HMA_PERIODS elif mode == 'LOWESS': PERIODS = LOWESS_PERIODS else: assert(False) for period in PERIODS: # --- initialisation - in preparation to fit iteratively ydf = pd.DataFrame([df['TPP L/NP'] / 100.0, df['Day'], df['Firm']], index=['y', 'Day', 'Firm']).T H = pd.get_dummies(df['Firm']) # House Effects dummy var matrix Houses = H.columns H = H.as_matrix() # --- undertake the analysis ... (iter_count, history, y) = curve_fit(Houses, H, mode, period, ydf, n_days) # --- record the minima generating house effects minima = get_minima(history) (iter_num, effects) = get_details(minima, history, Houses) ydf['adjusted_y'] = y + dot(H, effects) hidden_state = estimate_hidden_states(ydf, mode, period, n_days) print(type(hidden_state.index)) Hidden_States['{}-{}'.format(mode, period)] = hidden_state sum = sum_error_squared(hidden_state, ydf) effects = calculate_house_effects(Houses, ydf, hidden_state) house_effects = note_house_effects(effects, Houses, mode, period, iter_count, sum) Adjustments = Adjustments.append(house_effects) # get back to something useful hidden_state *= 100.0 hidden_state.index = [(start + x).to_timestamp().date() for x in hidden_state.index] fax = chart(sdf=hidden_state, title='{}-{}: Coalition TPP Vote Share'.format(mode, period), y_label='Percent Coalition TPP Vote', file_prefix=graph_leader+'{}-{}'.format(mode, period)) fax = mg_min_max_end(series=hidden_state, fax=fax, filename=graph_dir+graph_leader+ '{}-{} !annotated.png'.format(mode, period)) plt.close() print('\n-- FOUND --\n', house_effects, '\n-----------\n') # --- get an Adjustments summary in pro-Coalition percentage points Adjustments.index = Adjustments['Model'] AdjustmentsX = Adjustments[Houses] * -100 # Note: bias = -treatment * 100% #AdjustmentsX['Total'] = AdjustmentsX.sum(axis=1) AdjustmentsX['Iter'] = Adjustments['Iterations'] AdjustmentsX['Sum Errors Squared'] = Adjustments['Error Sq Sum'] print(AdjustmentsX.to_html()) print(AdjustmentsX) # --- Plot Hidden_States *= 100.0 # from proportions back to percent Bayes_TPP = pd.read_csv(intermediate_data_dir+ 'STAN-TPP-ZERO-SUM-walk.csv', header=0, index_col=0, quotechar='"', sep=',', na_values = ['na', '-', '.', '']) Hidden_States['Bayes'] = Bayes_TPP['median'] Hidden_States.index = [(start + x).to_timestamp().date() for x in Hidden_States.index] # allow us to annotate the end points endpoints = Hidden_States[-1:].copy().round(1) endpoints = 'Endpoints: ' + '; '.join([x+': '+str(y)+'%' for x,y in zip(endpoints.columns, endpoints[0:1].values[0])]) # and plot ... fig, ax = chart(sdf=Hidden_States, title='Coalition TPP Poll Aggregates [Sum(HE)==0]', y_label='Percent Coalition TPP Vote', annotation=endpoints, file_prefix=graph_leader+'!Comparative') plt.close() combination = ', '.join(Hidden_States.columns.values) Hidden_States['Average'] = Hidden_States.mean(axis=1) fax = chart(sdf=Hidden_States['Average'], title='Combined Coalition Aggregate TPP Vote Share', y_label='Percent Coalition TPP Vote', annotation=combination, file_prefix=graph_leader+'!Comparative !combined') fax = mg_min_max_end(series=Hidden_States['Average'], fax=fax, filename=graph_dir+graph_leader+'!Comparative !combined !annotated.png') plt.close()